Quantitative Techniques II

• What are your expectations from this course?

• My expectations

• Examples of optimization problems you’ve seen earlier?

• Examples of Mathematical modelling you’ve seen earlier?

• …here, in term1?

• E.g. in cases?

• Course Outline and Evaluation

Additional References

1. Applied Mathematical Programming

by Bradley, Hax, and Magnanti,

Addison-Wesley, 1977,

http://web.mit.edu/15.053/www/

2. Proceedings of the Advanced Workshop and Tutorial on Operations Research (AWTOR) 2012,

Organized by and at IIM Indore,

Sponsored by ORSI Ahmedabad Chapter,

Edited by Nagarajan K, and N Ravichandran,

Allied Publishers, 2014.

Example 1 (Source: TT Narendran, IIT Madras, “Mathematical Programming Models”,

Proceedings of AWTOR 2012)

A senior executive of a company recently optedfor VRS with a hefty packet of Rs. 5 crores. A ChitFund Company has offered the followinginvestment scheme for the benefit of suchretired people:

• "Invest a certain sum (in lakhs of rupees) atthe beginning of any month, invest half of thatamount beginning of the next month and endof the second month, you will get twice theamount invested originally in the first month".

• This scheme is available for the next sixmonths.

• The returns received at the end of any monthcan be used immediately for reinvesting eitheras a fresh investment or as a follow-upinvestment.

If you are in his/ her position, what will your aim be?

How will you achieve your aim/ objective?

Example 2(Source: Bradley, Hax, Magnanti)

A portfolio manager in charge of a bankportfolio has $10 million to invest. The securitiesavailable for purchase, as well as their respectivequality ratings, maturities, and yields, are shownin the following table:

Bond name

Bond type Quality Years to maturity

Yield to maturity

After-tax yield

A Municipal 2 9 4.3 4.3

B Agency 2 15 5.4 2.7

C Government 1 4 5.0 2.5

D Government 1 3 4.4 2.2

E Municipal 5 2 4.5 4.5

The bank places the following policy limitations on the portfolio manager’s actions:

1. Government and agency bonds must total at least $4 million.

2. The average quality of the portfolio cannot exceed 1.4 on the bank’s quality scale. (Note that a low number on this scale means a high-quality bond.)

3. The average years to maturity of the portfolio must not exceed 5 years.

If you are the portfolio manager, what would your objective be?

What all factors limit your objective?

(What are the constraints?)

• Mathematical Models:

– Deterministic

• Linear

• Non-linear

– Stochastic

• Linear Programming Problems (LPP) are a subclass of (deterministic) optimization problems.

• Is f = xy + yz a linear function?

• f(x,z) = xy + yz?

• f(y) = xy + yz is also linear.

• f(x, y, z) = xy + yz is not linear.

• Given that x and y are decision variables, is(x/y) + (x2/y) < 10x a linear constraint?

• To model the above examples:

– Decision variables

– Objective function

– Constraints

• Formulate examples 1 and 2 as LPPs.

• Some Mathematical basics

Questions?

Thank You